Theorem r19.43 1765

Description: Restricted version of Theorem 19.43 of [Margaris] p. 90.

Assertion

Ref	Expression
r19.43	|- (E.x e. A (ph \/ ps) <-> (E.x e. A ph \/ E.x e. A ps))

Proof of Theorem r19.43

Step	Hyp	Ref	Expression
1		andi 604	. . . 4 |- ((x e. A /\ (ph \/ ps)) <-> ((x e. A /\ ph) \/ (x e. A /\ ps)))
2	1	exbii 1051	. . 3 |- (E.x(x e. A /\ (ph \/ ps)) <-> E.x((x e. A /\ ph) \/ (x e. A /\ ps)))
3		19.43 1088	. . 3 |- (E.x((x e. A /\ ph) \/ (x e. A /\ ps)) <-> (E.x(x e. A /\ ph) \/ E.x(x e. A /\ ps)))
4	2, 3	bitr 173	. 2 |- (E.x(x e. A /\ (ph \/ ps)) <-> (E.x(x e. A /\ ph) \/ E.x(x e. A /\ ps)))
5		df-rex 1650	. 2 |- (E.x e. A (ph \/ ps) <-> E.x(x e. A /\ (ph \/ ps)))
6		df-rex 1650	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
7		df-rex 1650	. . 3 |- (E.x e. A ps <-> E.x(x e. A /\ ps))
8	6, 7	orbi12i 257	. 2 |- ((E.x e. A ph \/ E.x e. A ps) <-> (E.x(x e. A /\ ph) \/ E.x(x e. A /\ ps)))
9	4, 5, 8	3bitr4 183	1 |- (E.x e. A (ph \/ ps) <-> (E.x e. A ph \/ E.x e. A ps))

Syntax hints:   <-> wb 146   \/ wo 222   /\ wa 223   e. wcel 958  E.wex 980  E.wrex 1646

This theorem is referenced by:  r19.44av 1766  r19.45av 1767  r19.45zv 2352  iunun 2613  ssxr 5540

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-4 973  ax-5o 975

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-rex 1650

Copyright terms: Public domain